The Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Working Paper 263
December 1984
The Role of Intensional and
Extensional Representations
in Simulation
Daniel Brotsky
Abstract
I review three systems which do simulation in different domains. I observe the
following commonality in the representations underlying the simulations:
a
The representations used for individuals tend to be domain-dependent. These
representations are highly structured, concentrating in one place all the information concerning any particular individual. I call these representations
intensional because two such representations are considered equal if their
forms are identical.
* With important exceptions, the representations used for classes of individuals tend to be domain-independent. These representations are unstructured
sets of predications involving the characteristics of class members. I call these
representations extensional because two such representations are considered
equal if the classes they specify are identical.
I draw out various ramifications of this dichotomy, and speculate as to its cause.
In conclusion, I suggest research into the process of debugging extensional class
representations and the development of intensional ones.
This paper was prepared as the author's area examination.
A.I Laboratory Working Papers are produced for internal circulation, and may contain
information that is, for example, too preliminary or too detailed for formal publication.
It is not intended that they should be considered papers to which reference can be made
in the literature.
The Role of Intensional and
Extensional Representations
in Simulation
A task central to much of intelligent activity is simulation of real-world
processes. We are often forced to internally simulate external processes
in order to plan a course of action, predict the ramifications of events,
or analyze what went wrong. If we wish computers to perform planning,
prediction, or debugging tasks, we must develop algorithms and representations that enable them to do simulation. I consider here some of the
representational questions involved.
My comments in this paper arise from the study of three research efforts involving simulation. Davis (1984) simulates digital circuit operation,
Simmons (1983) simulates geologic processes, and Weld (1984) simulates
enzymatic interactions. All three authors choose their representations for
objects based on domain considerations, but the results seem to me to
display an interesting commonality. In particular:
*
*
The representations used for individuals tend to be domain-dependent.
These representations are highly structured, concentrating in one place
all the information concerning any particular individual. I call these
representations intensionalbecause two such representations are considered equal if their forms are identical.
With important exceptions, the representations used for classes of individuals tend to be domain-independent. These representations are
unstructured sets of predications involving the characteristics of class
members. I call these representations extensional because two such
representations are considered equal if the classes they specify are
identical.
This dichotomy in object representation leads to a similar, predictable dichotomy in process representation:
*
*
If a process operates on individuals, its representation tends to be
operational: it is specified as a series of additions to, deletions from,
and alterations of the representations of its targets.
If a process operates on classes, its representation tends to be declarative: it is specified as a set of predications about members of the class
which are asserted to be true when the operation completes.
I will comment at length a bit later on the probable source and ramifications
of this dichotomy in representation. But first I will spend some time making
the dichotomy concrete by examining two of the simulations.
Intensional and Eztensional Representations in Simulation
Simulation of Geologic Processes
Simmons (1983) addresses the task of recovering the geologic histories of
rock formations. He takes a diagram showing a rock formation, such as that
in figure 1, and hypothesizes a sequence of parameterized geologic processes
that would produce the pictured formation. For example, figure 2 shows a
sequence of processes that produce the formation of figure 1.
Simmons's system, only portions of which are implemented, approaches
this problem much as a geologist might. The system starts by noticing features of the formation that suggest the occurence of particular processes,
such as the igneous "dike" in figure 1. A temporal partial order is imposed
on the suggested processes by other features, such as the relative heights of
deposited layers. The end result of all the pattern matching is a partiallyordered sequence of geologic processes that together should account for
all the formation's features. The system then checks this sequence with a
simulation.
The verification process, termed imaginingby Simmons, has two goals:
*
*
Determine the physical feasibility of the proposed sequence.
Determine the quantitative parameters of each process.
Simmons achieves these goals by performing two simulations: one qualitative, one quantitative. The function of the qualitative simulation is to
determine process feasiblity and the interaction of quantitative parameters
among the various processes. This then allows the system to deduce exact
quantitative process parameters from measurements in the goal diagram.
The function of the quantitative simulation is to determine process feasibility and to verify that the hypothesized processes, when acting with the
hypothesized parameters, does indeed produce the goal formation.
Multiple Representations Support
Multiple Reasoning Techniques
The representations underlying the two simulations are a major focus of
Simmons's work. He uses a variety of distinct representation schemes for
different objects, justifying each in terms of the operations it must support.
For example:
*
*
Rock formations in the qualitative simulation are represented by framelike objects whose slots contain time lines of values. The time lines
facilitate reasoning about change over time.
Rock formations in the quantitative simulation are represented by diagram objects which are minimally-connected edge graphs. Each edge
Intensional and Extensional Representations in Simulation
*
is annotated with the faces it separates. This structure allows easy
measurement of diagram parameters and the efficient manipulation of
edges representing rock boundaries.
Qualitative and quantitative values used in the two simulations are
compared with each other using a partial-order lattice maintained by
specialized interval-arithmetic and truth-maintenance procedures.
Each of these representations is intensional. Simmons makes a careful analysis of the types of reasoning required by the domain, and then constructs
representations whose structure facilitates this reasoning. Questions about
instances are answered by checking local structure; changes in instance
properties are effected with local structural changes.
Unfortunately, Simmons does not explain how he keeps the various
representations consistent with each other-a task that could be quite difficult in general. As an aside, I conjecture that the issues involved in this
task are quite similar to those involved in modelling classes: one must understand all the paths of interaction between the various facets of instances
being represented separately. This question deserves further research.*
Different Representations Underly the Different Simulations
Simmons's quantitative simulation is completely supported by his intensional instance representations. Quantitative processes are represented as
sequences of diagram manipulations. For example, figure 3 shows Simmons's quantitative process model for erosion.
Simmons's qualitative simulation is not completely supported by intensional representations. Consider the qualitative process model of erosion
shown in figure 4. Portions of the model are specifications of intensional
operations; for example, the statement
(change = BA.orientation 0.0 I EROSION)
sets the orientation of the eroded surface to horizontal and marks EROSION
as the resposible process. Much of the model, however, is extensional:
statements such as
(> SURFACE.top.height@Istar t SEA-LEVEL)
(> ELEVEL SEA-LEVEL)
(< SURFACE.bottom.heightQIen d SURFACE.bottom.height§Istart )
make no alterations to the intensional representations. They serve instead
as predications on the possible rock formations that could be modelled by
*Perhaps some poor soul could do an area exam about it?
Intensional and Extensional Representations in Simulation
the intensional representations; that is, they limit the extensions of those
representations, *
Qualitative Rock Formations are Classes
of Quantitative Rock Formations
I claim that the distinction between the quantitative and qualitative process models arise because of a class/instance distinction in their target
representations. Simmons has perspicuous, intensional representations for
each individual rock formation he deals with, but he has no intensional representation for classes of formations. He must specify classes extensionally
by predicating appropriate characteristics for its members. This forces his
qualitative process representation into a declarative form:
*
*
Each process uses a set of (primarily geometric) preconditions to define
a class of instances to which it is applicable.
Each process uses a set of (primarily geometric) effects to define a
class of instances that contains the rock formation resulting from the
process.
Much of the geometric information contained in Simmons's diagrammatic
instance representation is duplicated in the intensional representations used
by the qualitative simulator. This is partially motivated by the qualitative
simulator's need to model the temporal changes in various quantities, a
need the quantitative simulator does not share. But another motivation
comes from his modelling of classes extensionally:
* Many of the predicates that limit the extensions of the qualitative
representations refer to geometric information that is in the diagram
representations. Thus, this information must be replicated in data
structures accessible to the qualitative simulator.
A natural way to avoid this duplication of information would be to use
some form of "blurred" diagrammatic representation to intensionally represent classes of instances. The qualitative process models could then be
operational. Simmons addresses this issue from two perspectives, albeit
indirectly. First, he says "... how a formation will split due to faulting, or
what the shape of the formation will be after erosion are both difficult to
express qualitatively." t This suggests that a good intensional representation would be hard to find. Second, Simmons says "... qualitative models
*To bring home the unimportance of the form of these assertions, consider that
the order of their arguments has no effect on their import; they could all have
been written using the complementary predicate.
tSimmons (1983), p. 25
Intensional and ExtenaionalRepresentations in Simulation
are inherently ambiguous, in that a single qualitative representation maps
to many real-world situations."* Since Simmons intends his diagrammatic
representation to be geometrically unambiguous, this statement suggests
that it might be methodologically incorrect to combine the class and instance representations.
I believe that Simmons is both right and wrong. In particular:
*
*
The development of good intensional representations for classes always
follows their development for individuals. Class representation development is guided by observation of instance behavior under controlled
circumstances-i. e., simulation-and such simulation can not proceed
without good representations for the individuals.
Whether intensional representations of classes and individuals should
be structurally related depends on the operations these representations
must support.t In Simmons's case, for example, the fact that both the
qualitative and quantitative processes manipulate geometic characteristics of individuals suggests that structurally similar representations
might be indicated.
I will return to these issues after an examination of Weld's simulation of
enzymatic interaction.
Simulation of Genetic Activity
Weld (1984) simulates the interactions of enzymes and proteins in genetic
activity. His system is given a description of the various proteins present
in a situation, together with process descriptions that govern their possible interactions, and predicts the stable state-if any-that the protein
interactions will lead to.
Weld's system, like Simmons's, performs two types of simulation in
order to make its predictions. First, it simulates protein interactions as
discrete operations involving a small number of proteins. As it performs
this simulation, it monitors the particular protein molecules involved in
each reaction. Then, when a cycle of reactions involving the same types
of proteins is noticed, the system suspends the discrete simulation. The
system transforms the representation of the discovered cycle from a discrete model into a so-called continuous model. In this view, the cycle is
considered an individual reaction which is continuously active, and Weld's
system simulates its continuous behavior in order to determine the limit
towards which it tends. This limiting state is then translated back into the
*Ibid, p. 25
tThis is a truism, of course, but worth repeating in this context.
Intensional and Extensional Representations in Simulation
discrete view and the first simulation starts again. A succinct example of
system behavior is given in Weld (1984), §1.1, page 2.
Discrete vs. Continuous is like Qualitative vs. Quantitative
I focus here on the representations that underly the two simulations.* As
with Simmons's work, we find that one of the simulations-the continuous one-rests on a base of entirely intensional representations, while the
other-the discrete simulation-manipulates intensionally represented individuals whose possible extensions are restricted by predications:
*
Each state of the world in the continuous model consists of a single reaction represented by a list of the amounts of surrounding proteins and
the influence (positive or negative) the reaction has on those proteins.
*
Each state of the world in the discrete model consists of molecular
clusters of various types (represented intensionally) which have history
(represented intensionally) and which are constrained extensionally to
be in certain relations to each other.
The discrete and continuous process models show the effects of the underlying dichotomy in representation, just as they did in Simmons's work:
*
A continuous processt is represented as an algorithm that alters the
representation of a continuous reaction according to the contents of
that representation and pre-computed limit points.
*
A discrete process is represented as a set of extensional preconditions
and a set of extensional effects, together with some intensional actions
on the underlying molecular representations. (The intensional actions
are encoded as demons which are attached to predicate names.) A
discrete process representation is shown in figure 5.
*Note that Weld's focus is neither the discrete nor the continuous simulations
in and of themselves, but the aggregation process that recognizes discreteprocess cycles and transforms them into continuous-process individuals. Thus,
he does not carefully justify his simulation representations to the extent that
Simmons does. In his literature review, he attributes much of his discrete-
simulation representation to work on situation-action rules (Fikes 71), and his
continuous-process simulation to work on Qualitative Process theory (Forbus
1983).
tThere is only one continuous process-the one that takes a reaction and computes its limit. It appears from Weld (1984) that this one is hard coded in
LISP.
Intensional and Eztensional Representations in Simulation
Continuous Individuals Represent Classes
of Discrete Individuals
I return now to the issue of using intentional class representations that I
raised at the end of my review of Simmons's work. Weld's work provides
an interesting perspective once we notice that his continuous-process individuals are in fact representations of classes of discrete-process cycles. For
example, consider the two sequences of reactions outlined in figure 6. In
the left-hand reaction, A causes E to fold so that it can catalyze the reaction
C " D; In the right-hand column it is B that does so. Since the cycles have
no net influence on either A or B, their continuous images are identical.
*
*
The continuous process individuals function as class representativesin
that they encapsulate in an intensional representation all the interesting characteristics of a class of discrete process cycles. The perspicuous
nature of this representation allows the efficient computation of relevant characteristics of the class, in Weld's case its limit point.
The function of Weld's aggregator is to identify process cycles and,
having done so, to construct the class representative for that cycle's
class.
This perspective suggests a research direction that Weld does not address,
but which speaks directly to the extensional/intensional dichotomy:
* Are there classes of non-cyclic discrete process chains that are of interest? Is there a perspicuous intensional representation for the class
representatives of such classes? If the discrete process simulator were
built on such a representation, would the aggregation task become
easier?
Causes of the Intensional/Extensional Dichotomy
I have argued that both Simmons and Weld use intensional representations for individuals and extensional ones for classes. As one might guess
from the questions.I have raised, I would prefer to see only intensional
representations used. My view is that:
*
*
We use extensional representations for classes because we do not know
how to characterize them completely. The extensional representations
allow us to assert what we know about the class without specifying it
completely.
Once we know of a complete characterization for classes, we use it
to construct intensional representations. Even when we only have
characterizations for some classes of interest, we construct intensional
representations for those classes.
Intensional and-Eztenaional Repreaentationa in Simulation
A good example of the use of intensional representation for distinguished
classes and extensional representation for others is provided by Davis (1984),
which I now examine.
Class Representatives in Circuit Simulation
Davis (1984) considers a variety of questions pertaining to the debugging
of digital circuits.* He presents a simulation technique that predicts the
behavior of an individual circuit using a highly intentional representation
built from the circuit's schematic. The circuit representation models each
component of the circuit as a small constraint network and then connects
these networks according to the schematic to produce a large constraint
network representing the circuit as a whole.t
Note that this constraint-network interconnection technique produces
an intensional representation for a single circuit, that is, an individual.
Thus, its utility is limited. For example, consider the circuit shown in
figure 7, and suppose that we suspect that component Add-1 is the single
element at fault. We could try to test this conjecture by enumerating likely
faults, constructing intensional representations for each resulting circuit,
and then simulating them to see if they produce the observed behavior.
But we would be better off using a simulation technique that operates on
classes of circuits at once.
A class simulator could be constructed using the sorts of extensional
techniques described above. But Davis presents a technique called constraint suspension that models classes of circuits using an intensional, constraint-network representation like that used for individual circuits. This
technique takes the constraint-network normally constructed for the circuit of figure 7 and locally modifies it to produce a network representing
the class of circuits with the same schematic whose Add-1 component is
faulty. The resulting network can then be used in a simple simulation that
determines whether there could be some member of the represented class
which displays the observed behavior.
Note that Davis does not seem to use intensional representations for
all classes of faulty circuits. For example, when speaking of identifying
bridge faults, he uses the following highly extensional language:
It is plausible to hypothesize a bridge fault between two modules A
and B from two different tests if: in test 1, A produced an erroneous 0
*Iwill not consider the large portions of Davis (1084) that address issues that
I am not concerned with in this paper.
tDavis makes a distinction between "forward" and "backward" constraint propagation that does not concern us here.
Intensional and Extensional Representations in Simulation
and B produced a valid 0, and in test 2, A produced a valid 0 while B
produced an erroneous 0.*
But the point here is that Davis gets a tremendous amount of leverage on
the simulation problem just by using intensional representations for wellunderstood classes of interest: in this case circuits with a single faulty
component.
Summary and Conclusions
I have reviewed the use of representations in a variety of simulations. I
have noticed the following commonality:
*
*
Individuals tend .to have domain-dependent, intensional representations. Operations on individuals are represented as algorithms that
manipulate the representations of their targets.
Classes tend to have domain-independent, extensional representations.
Operations on classes are represented as declarations in the group
representation language that modify the extension of the group.
I have claimed that extensional class representations are used for want
of the domain understanding necessary to produce intensional representations, and I have given examples from the reviewed papers of the use
and advantages of the occasional intensional class representation. I believe
that, to a large extent, representations in simulation systems evolve from
extensional to intensional:
*
*
*
*
When a domain is first explored, its objects and processes are not
well understood. Observed characteristics of objects and processes are
listed and used to generate extensional representations for both.
As further study reveals patterns unifying various observed characteristics, the extensional representations of individual domain objects are
supplanted by intensional representations which reveal the structure of
the objects they model. This allows the use of operational definitions
for processes.
Good representations for individuals and operations allow exact simulation, which facilitates work on planning and prediction in the domain. These tasks require classification of individuals in a manner
related to the domains and ranges of operations. Initial observations
about natural classes gives rise to an extensional class representation.
Extensive experience with simulation reveals the nature of interesting
classes of individuals, leading to intensional representations for these
classes.
*Davis (1984), p. 28.
Intensional and Extensional Representations in Simulation
The simulations reviewed here are all in domains whose individuals are well
understood but whose classes in general are not. In fact, the papers display
a range of levels of understanding of classes: circuit classes are understood
most clearly, then enzymatic reaction classes, and finally classes of rock
formations. Interestingly, one of the main concerns of both Weld's and
Davis's research is how to do effective prediction by making the best use
of a single well-understood class. This concern is predictable given the
evolutionary outline just presented.
Future Research
I want to close by mentioning a line of research suggested by the perspective
I have take here. I claim that extensional representations are used for
classes because they allow the use of partial specifications, particularly
those based on observations of hitherto unrelated characteristics. But a
variety of extensional representations are available, such as the various
logics, rule-based expert systems, and collections of heuristics. Which one
shall we use?
To answer this question, I believe we need to examine the process
by which we debug our extensional definitions. Both Simmons and Weld
describe errors in their simulations that arise from incorrect classifications.
We would like the representation system we use to give us as much help as
possible correcting such errors.
Unfortunately, the nature of this process is not very well understood,
nor are the techniques that can be used. Many cases of such debugging
are documented; for example, Simmons mentions that instances where his
qualitative and quantitative simulations disagree are helpful in correcting
his qualitative process definitions, and much literature exists about debugging rule-based expert systems. But so far no reviews have been done that
focus on the debugging of extensional classifications,especially as it relates
to simulation problems. Work in this area is needed before the basis for
choosing a representation can be anything other than esthetic.
Intensional and Extensional Representations in Simulation
SHALE
D.SANDSTONE
D MAFIC-IGNEOUS
Figure 1. A rock-formation diagram such as that interpreted by Simmon's geology system. The roughly-vertical slice of igneous rock pierc-:
ing both the shale and sandstone formations suggests an intrusion process. The fact that the sandstone is under the shale indicates that
the sandstone was deposited first, the shale second. [Illustration from
Simmons (1983).]
1. Deposit sandstone.
2. Deposit shale.
3. Uplift.
4. Intrude maf ic-igneous through sandstone and shale.
5. Tilt.
6. Fault.
7. Erode shale and mafic-igneous.
Figure 2. A sequence of geologic processes which would produce the
formation of figure 1. This sequence does not give the quantitative
parameters, such as width of intrusion and angle of tilt, which are
deduced by Simmons's system. [Illustration adapted from Simmons
(1983).]
Intensional and Extensional Representations in Simulation
Erosion (Elevel,. Eboundary)
1. Draw a horizonal line at Elevel.
2. Erase all parts of the line which do not cut across a face
corresponding to a rock-unit.
3. Erase all faces above the horizontal line.
4. The edges corresponding to SURFACE-the surface of
the Earth-are all the old SURFACE edges which were not
erased, plus the remaining edges of the horizontal line.
5. The edges corresponding to Eboundary-the erosional
boundary-are the remaining edges of the horizontal line.
Figure 3. Simmons's quantitative process definition of erosion. The
intensionality of the underlying diagram representation allows this definition to be completely operational: it works by altering the structure
of the underlying object representation. [Illustration adapted from Sim-
mons (1983).]
Intensional and Extensional Repreaentations in Simulation
EROSION
INTERVAL
I : temporal-interval
PRECONDITIONS (> SURFACE.top.height@Istar t SEA-LEVEL)
PARAMETERS
ELEVEL : real
C-PART
(set-of rock-unit), C-ALL : (set-of rock-unit).
AFFECTED .
SURFACE
CREATED
BA : boundary
EFFECTS
(change = BA.side-1 (THE-AIR) I EROSION)
(change = BA.side-2 C-PART I EROSION)
(change = BA.orientation 0.0 I EROSION)
(change = SURFACE.orientation
(efn3 ELEVEL SURFACE@Istart) I EROSION)
RELATIONS
(change function SURFACE.top (efn2 ELEVEL) I EROSION)
(change function SURFACE.bottom (efn2 ELEVEL) I EROSION)
(change function SURFACE.location (efn2 ELEVEL) I EROSION)
(for-all cl E C-PART
(and (change function cl.top (efnZ ELEVEL) I EROSION)
(change function cl.location (efn2 ELEVEL) I EROSION)
(change - cl.thickness
ELEVEL))
(* (efnl cl ELEVEL) (- cl.top.height@Ita
t
I EROSION))
(for-all c2 E C-ALL (change =.c2 1. I EROSION))
(> ELEVEL SEA-LEVEL)
(= C-PART (r : rock-unit
(and (exists-at r Ista)
(Q r.top.height@Ister t ELEVEL)
(< r.bottom.height@Istart ELEVEL)))})
(= C-ALL {r : rock-unit I
(and (exists-at r Istart)
(Q r.bottom.height@Itar t ELEVEL))}) .
(= (• SURFACE.bottom.height@Istar t ELEVEL)
(= SURFACE.bottom.height@Ien d SURFACE.top.height@Iend))
(= SURFACE.top.height@Iend ELEVEL)
( _ SURFACE.bottomaheight
Iend SURFACE.bottom.height@Istart)
(for-all cl E C-PART
(and (= cl.top.height@Iend ELEVEL)
(> (efnl ci ELEVEL) 0.0)
(< (efnl cl ELEVEL) 1.0)))
Figure 4. Simmons's qualitative process definition of erosion. The EFFECTS forms cause changes to the intensional rock-formation representations, while the RELATIONS forms are predications on those representations. Intuitively, the predications define the set of rock formations
which could result from the erosion process; that is, they limit the
possible extensions of the intensional representation.
Intensional and Eztensional Representations in Simulation
(BIND ?binder>?site ?bindee>?handle)
(and (gt? (number-of ?binder) zero)
(gt? (number-of ?bindee) zero)
(not (bound? ?binder>?site ?moleculel))
(not (bound? ?molecule2>?site2 ?bindee>?handle))
(complementary? ?binder>?site ?bindee>?handle))
(bound? ?binder>?site ?bindee>?handle)
Figure 5. A discrete process representation from Weld (1984). The first
form is a pattern used to bind the variables--those symbols starting
with a question mark. The first predicate is a precondition on the
state of the discrete world. The second predicate is an effect which is
asserted when the operation completes. The assertion of the effect has
intensional side effects in addition to its extensional effect.
Reaction 1
Reaction 2
(BIND E A)
(FOLD E)
(BIND E C)
(REACT C D)
(BIND E B)
(FOLD E)
(BIND E C)
(REACT C D)
(DROP E D)
(DROP E D)
(DROP E A)
(FOLD E)
(DROP E A)
(FOLD E)
repeat
repeat
Figure 6. Two discrete cycles which are aggregated to the same continuous individual. In the left-hand column, A causes E to fold so that it
can catalyze the reaction C H D; in the right-hand column it is B which
does so. Since the cycles have no net influence on either A or B, their
continuous images are identical.
Intensional and Eztensional Representations in Simulation
3
(12)
2
(10]
2
3
(12)
[12]
3
Actual---[]
A digital circuit. The expected output of Add-i differs from the actual
output, suggesting that Add-i is bad. Davis (1984) uses constraint
asupension to construct an intensional representation of the class of
circuits in which Add-i has a fault, and then does simulation using this
representation.